Quantitative weighted bounds for maximal singular integrals
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In this paper, we study weighted estimates for rough maximal singular integrals $T_\Omega^*$ which is defined by \begin{align*} T_{\Omega}^*f(x) &= \sup_{\varepsilon > 0} \left| \int_{|y| > \varepsilon} \frac{\Omega(y')}{|y|^n}f(x-y) dy \right|. \end{align*} More precisely, we first obtain the quantitative weighted bounds for \linebreak $\sup_{l \in \Bbb Z} |\phi_l \ast T_\Omega|$ where $T_\Omega$ is the singular integral operator with rough kernel. Secondly, we obtain quantitative $A_p$-weighted $(r' < p < \infty)$ bounds for $T_\Omega^*$ with $\Omega \in L^r(\Bbb S^{n-1})$ ($r > 1$), \begin{align*} \|T_\Omega^{*}f\|_{L^{p}(w)} \precsim \|\Omega\|_{L^r}[w]_{p/r'}^{\max\{1, \frac{1}{p-r'}\}}[w]_{p/r'}^{\max \{1, \frac{r'}{p-r'}\}} \|f\|_{L^p(w)}. \end{align*}
Abstract
In this paper, we study weighted estimates for rough maximal singular integrals $T_\Omega^*$ which is defined by \begin{align*} T_{\Omega}^*f(x) &= \sup_{\varepsilon > 0} \left| \int_{|y| > \varepsilon} \frac{\Omega(y')}{|y|^n}f(x-y) dy \right|. \end{align*} More precisely, we first obtain the quantitative weighted bounds for \linebreak $\sup_{l \in \Bbb Z} |\phi_l \ast T_\Omega|$ where $T_\Omega$ is the singular integral operator with rough kernel. Secondly, we obtain quantitative $A_p$-weighted $(r' < p < \infty)$ bounds for $T_\Omega^*$ with $\Omega \in L^r(\Bbb S^{n-1})$ ($r > 1$), \begin{align*} \|T_\Omega^{*}f\|_{L^{p}(w)} \precsim \|\Omega\|_{L^r}[w]_{p/r'}^{\max\{1, \frac{1}{p-r'}\}}[w]_{p/r'}^{\max \{1, \frac{r'}{p-r'}\}} \|f\|_{L^p(w)}. \end{align*}